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A simple trapezoid can self-tile the plane in many interesting periodic ways. This trapezoid is half of a regular hexagon. 
A couple more periodic tilings
Trapezoid tiling with 6-fold rotational symmetry. A Pattern Block set with one tile type, a dozen colors could be fun.
Trapezoids or "house" pentagons can self-tile similarly here.
A mostly "house pentagon" tiling, with some yellow trapezoids in the gaps
I suppose this would be too festive for a chess board, house-pentagons and isotoxal octagons.
Some more festive checkerboards
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